AC Quantities: Instantaneous

AC Quantities: Instantaneous

1. What Instantaneous Value Is

Instantaneous value $v(t), i(t)$ = value of alternating quantity at a specific instant $t$.

For sinusoidal AC (most common):

where $V_m$ = peak, $ω=2πf$, $φ$ = phase at $t=0$.

At $t=0$, $v(0)=V_m sinφ$; at $ωt=90°$, $v=V_m$.

2. Why Instantaneous Matters

• KVL/KCL apply instantaneously: $v_R(t)+v_L(t)+v_C(t)=v_s(t)$ at every $t$, even though $V_{rms}$ relations need phasors.
• Power instant: $p(t)=v(t)·i(t)$ pulsates at $2ω$ (even if $v,i$ at $ω$).

3. Worked Examples

(a) v(t)=325 sin(314t) V (mains): at t=5ms (quarter period, ωt=90°? Actually 314·0.005=1.57 rad=90°) ⇒ v=325V peak ✓
    At t=2.5ms (45°) ⇒ v=325·sin45°≈230V (which equals RMS? coincidence)

(b) i(t)=10 sin(314t+30°) A: at t=0, i=5A ✓

(c) Need to find instantaneous when to switch? At zero crossing vs peak matters for inrush.

4. Exam Points

• Define $v(t), i(t)$ and $V_m sin(ωt+φ)$ — 3-5 marks, very frequent.
• Show that $v(t)$ varies sinusoidally — 3 marks.
• Compute instantaneous at given $t$ — 5 marks, frequent.